1. OVERVIEW

In this section, I shall give a brief overview of the properties of
the Universe at large. On the largest scales it can be reasonably
approximated as a homogeneous and isotropic medium in a state of
uniform expansion and the equations can easily be written down. We
find that such a simple model Universe can be described in terms of a
few parameters, the expansion rate, the density, and perhaps the
cosmological constant. Classical cosmology focusses on determining
these by direct observation of the large scale distribution of
galaxies. There are, however, many new techniques available for
getting these parameters though studying the inhomogeneity of the
Universe. These will be the subject of the following sections where
many of the issues raised here will be discussed at greater length.

Hubble discovered the expansion of the Universe by plotting, for a
sample of galaxies, the radial velocity of each galaxy as indicated by
the redshift of its spectral lines against its apparent
brightness. The fainter (and presumably more distant) galaxies had the
greater recession velocities (or "redshifts"). If the distance to a
galaxy was D Megaparsecs, and its radial velocity was V km
s-1, then Hubble's relationship could be expressed as

(1)

where H0 is a constant (the Hubble constant)
measured here in units of
km s-1 Mpc-1 Implicit in the relationship is the
assumption
that we can calibrate the distance scale by virtue of which the apparent
brightness of a galaxy can be turned into a distance.

The radial component of the velocity of a galaxy relative to the
observer is inferred by observing the wavelength
0 of spectral
lines that would in the laboratory have been emitted wavelength
E. The
difference =
0 -
E is
interpreted as being due to the Doppler
shift caused by the fact that the galaxy was moving at velocity

(2)

relative to the observer. (We shall henceforth drop the `E' suffix on
the emitted wavelength). The redshift of the galaxy (in fact the
redshift of the spectra lines) is defined as

(3)

Hubble's redshift-distance relation (the "Hubble Law") later became
a way of estimating the distances to galaxies simply by measuring
their radial velocities
Dz = H0-1cz. (Dz has the subscript z to denote
the nature of this distance estimate and to distinguish it from the
true distance. We shall see later that part of the velocity cz may be
due to the random motions of galaxies relative to the general cosmic
expansion.)

Looked at in its most simple terms, Hubble's discovery implies that
the Universe was born a finite time in our past and emerged from a
state of infinite density. The subsequent discovery by
Penzias and Wilson (1965)
of a cosmic microwave background radiation field and its
interpretation as the relict of an expansion from a hot singular state by
Dicke, Peebles, Roll and
Wilkinson (1965)
established a definitive
view of our Universe. Cosmology properly became a branch of physics,
and the Hot Big Bang theory has become a paradigm of modern science.

On the smallest scales the Universe contains stars that are grouped
into galaxies, that are themselves grouped into clusters. Going to
larger scales we have evidence for clusters of galaxy clusters, and
beyond that for large scale structures ("walls" of galaxies!)
extending over many tens or even hundreds of megaparsecs. Indeed
pictures of the three dimensional distribution of galaxies look very
inhomogeneous even on scales as large as 100 Mpc, or more. However,
one should not be mislead by visual appearances. As will be explained
later, this large scale inhomogeneity has rather a small amplitude in
the sense that it would hardly be noticeable if the distribution of
galaxies were smoothed over such large volumes. There is a clear
tendency for the Universe to become more homogeneous on ever large
scales.

Hubble himself commented on the remarkable large-scale isotropy of
the Universe as judged from the distribution of galaxies on the
sky. Today we have catalogues of galaxies penetrating to great distances
(Maddox et al., 1990)
and these demonstrate the isotropy of
the galaxy distribution very clearly. The isotropy of the Universe is
best measured through the isotropy of the cosmic microwave background
radiation.

The large scale homogeneity of the Universe is more difficult to
establish directly. It would seem reasonable to use the argument that
we are not at the center of the Universe, so the isotropy must imply
spatial homogeneity, but this is not a proof of homogeneity. The same
deep galaxy catalogues provide a test of homogeneity because we can
ask the question "is the Universe, sampled at various depths within
this catalogue, the same?". Again the
Maddox et al. (1990)
catalogue
provides an answer, though the method is not as simple as observing
isotropy. Maddox et al. compute the galaxy clustering correlation
function at various depths in their catalogue and find that the
functions in the various samples scale in accordance with the
hypothesis of homogeneity. Their analysis in fact goes even further
than merely saying that the Universe is globally homogeneous. It has
the additional implication that the deviation from homogeneity (as
evidenced by the galaxy clustering) is itself the same in all their
samples.

Such arguments provide compelling evidence that the Universe is not
a hierarchy of the kind originally envisaged by
Charlier (1908,
1922),
and taken up more recently in the context of fractal distributions of
galaxies by
Mandelbrot (1983),
Coleman, Pietronero and
Sanders (1988)
and others.

For most of what concerns us in these lectures it is sufficient to
consider the Universe to be, in a first approximation, a homogeneous
and isotropic distribution of particles (galaxies) that interact only
through their mutual gravitational interactions. This means that we
ignore any pressure contribution from their random motions, or from
other components of matter. This enables us to greatly simplify the
dynamical equations for the evolution of the Universe.

Consider the motion of a galaxy in the Universe that today
(t0) is
at distance l0 from us and that at time t was
at a distance l (t). It
is convenient to define the scale factora(t) by

(4)

Since the Universe is presumed homogeneous and isotropic, then
a(t)
depends on neither position nor direction. It merely describes how
relative the distances change as the Universe expands. We have
normalized all lengths relative to their present day value and so the
present value of a(t) is
a(t0) = 1.

The Einstein equations (or their Newtonian equivalent) in the simple
case of homogeneous and isotropic dust models give the differential
equation for the scale factor in terms of the total mass density
:

(5)

This is supplemented by an equation expressing the conservation of
matter:

(6)

which is equivalent to

(7)

Note that (5) is not valid if there is any substantial pressure due to
the matter in the universe, and in that case we also need to modify
(6). We shall make these modifications at a later time when needed,
for the moment we are only discussing the Universe at the present time
and in its recent past when equations (5, 6, 7) are thought to be a
good approximation.

The Hubble Parameter is defined as

(8)

and is a function of time. H describes the rate of expansion of the
Universe and has units of inverse time. It is experimentally measured
as a velocity increment per unit distance since it describes the
expansion through the relationship between velocity and distance:
= H l, or in more
familiar notation v = H r.

We define the redshift to a galaxy at distance l to be

(9)

When we look at a distant galaxy we are looking at it as it was in the
past (because of the finite light travel time). At the time we are
seeing it, the scale factor a(t) was smaller than the
present value
(a0 = 1). It can easily be shown that the recession
velocity we
measure from the shift in the spectral lines is just cz, in other
words, the quantities z appearing in equations (3) and (9) are the
same thing.

At this point it is convenient to introduce some fundamental
definitions. Hubble's expansion law states that the recession velocity
of a galaxy is proportional to its distance from the observer, in
other words l0. The constant of proportionality (the
cosmic expansion rate) is the present value of the Hubble parameter:

(10)

H0, the present value of the Hubble Parameter, is
usually called "Hubble's Constant".

There is an important value of the density,
c,
that can be derived
from the Hubble parameter (the Hubble parameter has dimensions
[time]-1). This is the density such that a uniform
self-gravitating sphere of density
c
isotropically expanding at rate H has equal
kinetic and gravitational potential energies:

(11)

Since H is a function of time, then so is
c.

We can measure the density of the Universe in terms of
c by
introducing the density parameter:

(12)

Note that also
depends on time and we shall denote the present day
value of by
0. There
may be a mixture of different type of matter
in the universe that make up the total density
. We may think, for
example, of baryons, photons and perhaps some exotic elementary
particles. Each of these individually has a density that can be
normalized relative to
c,
thus each species has its own
. We will, for
example, denote the contribution of Baryonic
material to the total cosmic density by
B.

The density
c
has a special significance. A universe whose density
is c
when its expansion rate is H is referred to as an Einstein de
Sitter universe. This model clearly has
= 1 at all
times. The expansion rate of such a universe is fixed by the density. Model
universes that are denser than
c
= 3H2 / 8G when their expansion rate
is H will stop expanding and contract down to future
singularity. Models that are less dense will expand forever. The
= 1
universe is a limiting case dividing two classes of behaviour and that
is why the parametrization of the density in terms of
c is so
useful. The behaviour of the various model universes as a function of
can be seen by
looking at dynamical equation for the expansion factor a(t).

Equations (5) and (7) for a(t) can be shown to integrate to

(13)

The integration constants have been derived using the boundary
condition that a(t)
0 as
t 0,
( / a)0
= H0 and that the present density of matter is
0 =
0c.
The standard textbooks referred to
above give the solutions of this equation for general values of
0. It
is sufficient here to note that the case
0 = 1
simplifies the right hand side of this equation and the solution is then
particularly simple

Since
a(t) = (1 + z)-1 this tells us that when
we look back to a
redshift z in Einstein de Sitter universe we are seeing the universe
when its age is a fraction
t / t0 = (1 + z)-3/2 of its
present age, t0.